
Given we now have weight-shifting operators that both preserve and raise the spacetime dimension, is it also possible to construct operators that lower the spacetime dimension?
One approach we have explored, explained in appendix , is to find so-called Bernstein-Sato operators which act to lower the powers to which the various polynomials of interest are raised. In this case, the relevant polynomials are
the Cayley-Menger determinant and its minors appearing in the parametrisation (REF ).
We found, for example, that replacing \(v_{ij}\rightarrow \partial _{s_{ij}}\)  in the Kirchhoff polynomial \(\mathcal {U}=|g|\)  yields an operator
\(\mathcal {B}_{|m|} = (|g|)\Big |_{v_{ij}\rightarrow \partial _{s_{ij}}}\) 
which lowers by one the power to which the Cayley-Menger determinant is raised:
\(\mathcal {B}_{|m|}\, |m|^a = b_{|m|}(a) |m|^{a-1}, \qquad b_{|m|}(a) =- \prod _{k=1}^{n-1}(1-k-2a).\) 
For the simplex representation (REF ), \(a\)  is the parameter \(\alpha \)  given in (REF ) and so lowering \(\alpha \)  by one corresponds to sending \(d\rightarrow d-2\)  if all the operator dimensions are kept fixed. In principle, one would then integrate by parts to obtain an operator acting solely on the Schwinger exponential, which, due to its diagonal structure, could be
translated into a differential operator in the external momenta.
In practice, however, this approach
is complicated by the presence of all the
remaining powers of Cayley-Menger minors present in (REF ).

In sections  and REF , we saw how the action of the special conformal Ward identity on the simplex reduces to a total derivative. This followed directly from the scalar parametric representation, without any recourse to the recursive arguments developed in [1]}, [2]}.
Nevertheless, these arguments, and the recursion relation between \(n\) - and \((n+1)\) -point simplices on which they are based, are of considerable
interest in their own right and could be reformulated in the scalar-parametric language used here.
The deletion/contraction relations of graph polynomials (see, e.g., [3]}) and Kron reduction, corresponding to taking the Schur complement of a subset of vertices in the simplex Laplacian (see e.g., [4]}), may also yield relevant identities.

Starting from the general simplex solution, the arbitrary function of momentum-space cross ratios can be restricted by imposing additional conditions of interest: for example, dual conformal invariance [5]}, [6]}, [7]}, [8]}, or the Casimir equation for conformal blocks. For such investigations, the connection with position-space developed in section  provides a very simple link between the action of a given differential operator in the external momenta or coordinates, and its corresponding action on the arbitrary function of the simplex representation.

For holographic \(n\) -point functions, bulk scalar Witten diagrams have the interesting property that their form is invariant under the action of a shadow transform on any of the external legs. In momentum space, shadow transforming the operator \(\mathcal {O}_{i}\)  corresponds to multiplying the correlator by \(p_i^{-2\beta _i}\) , where \(\beta _i=\Delta _i-d/2\) , which has the effect of replacing \(\beta _i\rightarrow - \beta _i\)  in the bulk-boundary propagator
\(z^{d/2} p_i^{\beta _i} K_{\beta _i}(p_iz)\) .
It would be interesting to understand the restriction this condition places on the function of cross-ratios appearing in the simplex representation.

Finally, the parametric representations we have developed may provide a useful starting point for the construction of general spinning \(n\) -point correlators via the action of spin-raising operators [9]}, [10]}, [11]}.

